CAYLEY-BACHARACH SCHEMES AND THEIR CANONICAL MODULES

A set of s points in P(d) is called a Cayley-Bacharach scheme (CB-scheme), if every subset of s - 1 points has the same Hilbert function. We investigate the consequences of this ''weak uniformity.'' The main result characterizes CB-schemes in terms of the structure of the canonical module of their projective coordinate ring. From this we get that the Hilbert function of a CB-scheme X has to satisfy growth conditions which are only slightly weaker than the ones given by Harris and Eisenbud for points with the uniform position property. We also characterize CB-schemes in terms of the conductor of the projective coordinate ring in its integral closure and in terms of the forms of minimal degree passing through a linked set of points. Applications include efficient algorithms for checking whether a given set of points is a CB-scheme, results about generic hyperplane sections of arithmetically Cohen-Macaulay curves and inequalities for the Hilbert functions of Cohen-Macaulay domains.